Based on space-charge wave theory, the analytical expression of the beam-wave maximum bunching current in the multi-gap coupled cavity is derived. Through calculating maximum bunching current with different gap number and perveance for 2π mode, the bunching efficiency in the multi-gap coupled cavity is discussed.Extended-interaction klystron (EIK) has the advantages of traditional klystron's high reliability and high output power and can achieve a large power, wide frequency bandwidth and high gain in the high-frequency range by adopting the multigap coupled cavity.Based on space-charge theory, Chodorow and Wessel-Berg discussed the mechanism of the beam-wave synchronization and coupling in the multi-gap coupled cavity chain with extended interaction fields [1]. They found that the experimental performances accord well with the theoretical ones. However, their results were obtained based on a one-dimensional grid gap, and the effects of the space-charge parameter on the beam-wave interaction in the multi-gap coupled cavity chain were neglected. In this paper, according to space-charge wave theory, starting from Maxwell's equations, the normalized bunching current in the multi-gap coupled cavity can be written as:Where E c (ξ) is the axial electric filed intensity at a distance z, V 0 and I 0 are direct beam voltage and direct beam current, β q and β e are reduced plasma propagation constant and electron propagation constant, V n is the voltage on the n-th gap, l n, N is the distance between n-th gap and N-th gap, M n (β e -β q ) and M n (β e +β q ) are respectively the coupling coefficient for the fast and slow space-charge wave [2].Based on the equation (1), the normalized bunching current in the multi-gap coupled cavity can be calculated with the drift distance x. The velocity modulation occurs in the gap; the density modulation occurs mainly in the subsequent drift process and forms a maximum bunching current at a particular position. The bunching current can be recognized by the interference between the fast space-charge wave and the slow Figure 1. |i/I 0 |max vs. beam voltage (a) Per=0.5µp (b) Per=0.5µpspace-charge wave. Fig. 1 shows the curves of the normalized maximum bunching current |i/ I 0 | max versus the beam voltage with different gap number for 2π mode, where α is the modulation index of the gap voltage.When perveance is 0.5µp, the |i/I 0 | max moderately raises monotonically with the beam voltage at less gap number. With the increased gap number N, the normalized maximum bunching current |i/I 0 | max varies with beam voltage more rapidly and there is a maximum value at large N gap number. The magnitudes of these extremums enhance with the increased gap number N, and the corresponding voltage is close to the synchronization voltage 20 kV. Because the M n (β e -β q ) and M n (β e +β q ) are approximately equal with less gap number or β q <<β e , the maximum bunching current occurs at the synchronization voltage of coupling coefficient.When the perveance is 1.0µp, the force of space charge enhances wi...